Cross-intersecting families of vectors

Given a sequence of positive integers $p = (p_1, . . ., p_n)$, let $S_p$ denote the family of all sequences of positive integers $x = (x_1,...,x_n)$ such that $x_i \le p_i$ for all $i$. Two families of sequences (or vectors), $A,B \subseteq S_p$, are said to be $r$-cross-intersecting if no matter how we select $x \in A$ and $y \in B$, there are at least $r$ distinct indices $i$ such that $x_i = y_i$. We determine the maximum value of $|A|\cdot|B|$ over all pairs of $r$- cross-intersecting families and characterize the extremal pairs for $r \ge 1$, provided that $\min p_i >r+1$. The case $\min p_i \le r+1$ is quite different. For this case, we have a conjecture, which we can verify under additional assumptions. Our results generalize and strengthen several previous results by Berge, Frankl, F\"uredi, Livingston, Moon, and Tokushige, and answers a question of Zhang

On symmetric intersecting families of vectors

A family of vectors $A \subset [k]^n$ is said to be intersecting if any two elements of $A$ agree on at least one coordinate. We prove, for fixed $k \ge 3$, that the size of a symmetric intersecting subfamily of $[k]^n$ is $o(k^n)$, which is in stark contrast to the case of the Boolean hypercube (where $k =2$). Our main contribution addresses limitations of existing technology: while there is now some spectral machinery, developed by Ellis and the third author, to tackle extremal problems in set theory involving symmetry, this machinery relies crucially on the interplay between up-sets and biased product measures on the Boolean hypercube, features that are notably absent in the problem at hand; here, we describe a method for circumventing these barriers.Comment: 6 pages; It has been brought to our attention that our main result (with slightly worse estimates) may be deduced from earlier work of Dinur, Friedgut and Regev, and this revision acknowledges this fac

Invitation to intersection problems for finite sets

Extremal set theory is dealing with families, . F of subsets of an . n-element set. The usual problem is to determine or estimate the maximum possible size of . F, supposing that . F satisfies certain constraints. To limit the scope of this survey most of the constraints considered are of the following type: any . r subsets in . F have at least . t elements in common, all the sizes of pairwise intersections belong to a fixed set, . L of natural numbers, there are no . s pairwise disjoint subsets. Although many of these problems have a long history, their complete solutions remain elusive and pose a challenge to the interested reader.Most of the paper is devoted to sets, however certain extensions to other structures, in particular to vector spaces, integer sequences and permutations are mentioned as well. The last part of the paper gives a short glimpse of one of the very recent developments, the use of semidefinite programming to provide good upper bound

Cross-Intersecting Families of Vectors

Given a sequence of positive integers $$p=(p_1,\dots ,p_n)$$ p = ( p 1 , ⋯ , p n ) , let $$S_p$$ S p denote the family of all sequences of positive integers $$x=(x_1,\ldots ,x_n)$$ x = ( x 1 , ... , x n ) such that $$x_i\le p_i$$ x i ≤ p i for all $$i$$ i . Two families of sequences (or vectors), $$A,B\subseteq S_p$$ A , B ⊆ S p , are said to be $$r$$ r -cross-intersecting if no matter how we select $$x\in A$$ x ∈ A and $$y\in B$$ y ∈ B , there are at least $$r$$ r distinct indices $$i$$ i such that $$x_i=y_i$$ x i = y i . We determine the maximum value of $$|A|\cdot |B|$$ | A | · | B | over all pairs of $$r$$ r -cross-intersecting families and characterize the extremal pairs for $$r\ge 1$$ r ≥ 1 , provided that $$\min p_i>r+1$$ min p i > r + 1 . The case $$\min p_i\le r+1$$ min p i ≤ r + 1 is quite different. For this case, we have a conjecture, which we can verify under additional assumptions. Our results generalize and strengthen several previous results by Berge, Borg, Frankl, Füredi, Livingston, Moon, and Tokushige, and answers a question of Zhang